.. role:: blue
   :class: blue

MbRevolutionSurface Revolution Surface
=======================================
.. _MbRevolutionSurface_Revolution_Surface_:

.. версия ядра: 1180ХХ

The :blue:`MbRevolutionSurface` class is declared in the *surf\_revolution\_surface.h* file.

The :blue:`MbRevolutionSurface` revolution surface belongs to the group of
swept surfaces :blue:`MbSweptSurface`. The revolution surface is a special case
of a swept surface with a spine curve in the form of a circle or its arc. The
revolution surface is described by the generatrix curve
:ref:`MbCurve3D <MbCurve3D_Curve_>` **curve**, the local coordinate system
:ref:`MbPlacement3D <MbPlacement3D_Local_Coordinate_System_>` **position**, whose vector
**position.axisZ** is the axis of revolution, the Boolean flag
*planeData* indicating that the curve and the axis of revolution lie in one plane,
the Boolean flag *poleMin* indicating that the surface has a pole at the initial
value of the first parameter, the Boolean flag *poleMax* indicating that the surface
has a pole at the terminal value of the first parameter, and the values of the first
surface parameter at the poles of the surface *uPoleMin* and *uPoleMax*, if the
corresponding pole is present. The surface also has some additional data that are
optional and are used to speed up the execution of the surface methods.

The first surface parameter *u* coincides with the parameter of the generatrix
curve. The first surface parameter takes values on the interval
*umin* ≤ *u* ≤ *umax*, which corresponds to the domain
of the generatrix curve. The surface can be periodic with respect to
the first parameter if the generatrix curve is periodic.

The second surface parameter *v* takes values on the interval:
*vmin* ≤ *v* ≤ *vmax*. The values *v* = 0 and *v* = 2π
correspond to a point on the generatrix curve. The surface can be
periodic with respect to the second parameter. For a periodic surface,
*vmax* - *vmin* = 2π; for a non-periodic surface,
*vmax* - *vmin* < 2π.

In the :blue:`PointOn`\( double *u*, double *v*,
:ref:`MbCartPoint3D <MbCartPoint3D_Radius_Vector_of_Point_in_3D_Space_>` & **s** ) method, the position vector of the surface **s**
is described by the vector function

.. math::
   :nowrap:

   \begin{equation}
   \mathbf{s}(u, v) = \mathbf{position.origin} +
   \left( \mathbf{curve}(u) - \mathbf{position.origin} \right) \mathbf{M}(v),
   \end{equation}

where **M**\(*v*) is the rotation matrix. Note that
the vector (**curve**\(*u*) - **position.origin**)
is multiplied by the matrix **M**\(*v*) on the right. The rotation matrix
has the form

.. math::
   :nowrap:

   \begin{equation}
   M(v) = A^{-1} \cdot
   \begin{bmatrix}
   \cos v & -\sin v & 0 \\
   \sin v &  \cos v & 0 \\
   0      &  0      & 1
   \end{bmatrix}
   \cdot A =
   \begin{bmatrix}
   \mathbf{position.axisX} \\
   \mathbf{position.axisY} \\
   \mathbf{position.axisZ}
   \end{bmatrix}^{-1}
   \cdot
   \begin{bmatrix}
   \cos v & -\sin v & 0 \\
   \sin v &  \cos v & 0 \\
   0      &  0      & 1
   \end{bmatrix}
   \cdot
   \begin{bmatrix}
   \mathbf{position.axisX} \\
   \mathbf{position.axisY} \\
   \mathbf{position.axisZ}
   \end{bmatrix}.
   \end{equation}

The matrix **A** is the transformation matrix of the coordinates of the
position vector of a point from the local coordinate system **position** to the
global coordinate system. The rows of the matrix **A** are composed of the
components of the basis vectors of the local coordinate system. The matrix
**M**\(*v*) maps the vector **curve**\(*u*) - **position.origin**
to the local coordinate system, rotates it in this system by the angle *v* around the
axis of revolution, and maps the rotated vector back to the global coordinate system.
The revolution surface is shown in Fig. 1:

.. figure:: /MODELER/SURFACES/pictures/O.5.8.1.png
   :width: 12cm
   :align: center

   Fig. 1

If the initial or terminal edge of the generatrix curve passes through the axis
of revolution, then the surface has a pole for the parameter *umin* or
*umax*, respectively. The following inequality must hold for the boundary
values of the second surface parameter:
*vmin* < *vmax*.

The points of the generatrix curve are rotated along an arc around the vector
**position.axisZ** from the vector **position.axisX** in the direction of
the vector **position.axisY**. The local coordinate system
**position** can be either right-handed or left-handed.